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ICS (Intermediate in Computer Science) Mathematics (Part I) - Algebra and Trigonometry Syllabus

The Mathematics (Part I) - Algebra and Trigonometry syllabus for ICS (Intermediate in Computer Science) in full: 12 chapters and 44 topics, every one listed below, plus 52 Mathematics (Part I) - Algebra and Trigonometry flashcards with answers.

12Chapters

44Topics

52Flashcards

Mathematics (Part I) - Algebra and Trigonometry syllabus: all 12 chapters

Expand any chapter to see its topics and sub-topics.

Number Systems
3 topics
- Real Numbers and Their Properties
- Complex Numbers
- Properties of Real and Complex Numbers
Sets, Functions and Groups
4 topics
- Sets and Operations on Sets
- Venn Diagrams and Laws
- Relations and Functions
- Binary Operations and Groups
Matrices and Determinants
5 topics
- Types of Matrices
- Matrix Operations
- Determinants and Properties
- Inverse of a Matrix
- Solving Systems of Linear Equations
Quadratic Equations
4 topics
- Solution of Quadratic Equations
- Nature of the Roots
- Sum and Product of Roots
- Equations Reducible to Quadratic Form
Partial Fractions
3 topics
- Proper and Improper Fractions
- Resolution into Partial Fractions
- Linear and Quadratic Factors
Sequences and Series
4 topics
- Arithmetic Progression
- Geometric Progression
- Harmonic Progression
- Infinite Geometric Series
Permutation, Combination and Probability
4 topics
- Factorial Notation
- Permutations
- Combinations
- Basic Probability
Mathematical Induction and Binomial Theorem
4 topics
- Principle of Mathematical Induction
- Binomial Theorem for Positive Integral Index
- General Term and Middle Term
- Binomial Series
Fundamentals of Trigonometry
3 topics
- Units of Angular Measurement
- Trigonometric Ratios
- Trigonometric Identities
Trigonometric Identities of Sum and Difference of Angles
3 topics
- Sum and Difference Formulae
- Double and Half Angle Identities
- Product to Sum Formulae
Application of Trigonometry
4 topics
- Solution of Triangles
- Law of Sines and Cosines
- Area of a Triangle
- Circles Connected with Triangles
Inverse Trigonometric Functions and Trigonometric Equations
3 topics
- Graphs of Trigonometric Functions
- Inverse Trigonometric Functions
- Solving Trigonometric Equations

ICS (Intermediate in Computer Science) Mathematics (Part I) - Algebra and Trigonometry flashcards — 52 questions with answers

Define a rational number and give its standard form.
A number expressible as p/q where p and q are integers and q ≠ 0. In standard form q > 0 and HCF(p, q) = 1 (the fraction is in lowest terms).
What distinguishes a rational number from an irrational number in terms of decimal expansion?
A rational number has a terminating or recurring (repeating) decimal expansion; an irrational number has a non-terminating, non-recurring decimal expansion.
State the closure, commutative, associative, identity and inverse properties for addition of real numbers.
For all real a, b, c: closure (a+b is real), commutative (a+b=b+a), associative ((a+b)+c=a+(b+c)), additive identity 0 (a+0=a), additive inverse −a (a+(−a)=0).
State the distributive property of multiplication over addition for real numbers.
For all real a, b, c: a(b+c) = ab + ac (left distributive) and (b+c)a = ba + ca (right distributive).
State the trichotomy property of real numbers.
For any two real numbers a and b, exactly one of the following holds: a < b, a = b, or a > b.
State the transitive property of inequality for real numbers.
For real a, b, c: if a < b and b < c, then a < c (similarly for >, ≤, ≥).
What is the additive identity and the multiplicative identity in the real numbers?
The additive identity is 0 (a + 0 = a); the multiplicative identity is 1 (a · 1 = a).
Which real number has no multiplicative inverse, and why?
0 has no multiplicative inverse because there is no real number x such that 0 · x = 1.
Define a complex number and name its real and imaginary parts.
A complex number is z = a + bi where a, b are real and i = √(−1). Here a is the real part Re(z) and b is the imaginary part Im(z).
What is the value of i, i², i³ and i⁴?
i = √(−1), i² = −1, i³ = −i, i⁴ = 1. Powers of i cycle with period 4.
Define the conjugate of a complex number z = a + bi and state z·z̄.
The conjugate is z̄ = a − bi. Their product z·z̄ = a² + b² (a non-negative real number).
Define the modulus (absolute value) of a complex number z = a + bi.
|z| = √(a² + b²), the distance of the point (a, b) from the origin in the complex plane.
How do you divide complex numbers, e.g. (a+bi)/(c+di)?
Multiply numerator and denominator by the conjugate of the denominator: (a+bi)(c−di) / (c²+d²), then separate into real and imaginary parts.
Give the formula for multiplying two complex numbers (a+bi)(c+di).
(a+bi)(c+di) = (ac − bd) + (ad + bc)i.
When are two complex numbers a+bi and c+di equal?
They are equal if and only if their real parts are equal and their imaginary parts are equal: a = c and b = d.
How is a complex number z = a + bi represented geometrically?
As the point (a, b) in the Argand (complex) plane, with the x-axis as the real axis and the y-axis as the imaginary axis.
State the properties of the conjugate: conjugate of a sum and of a product.
Conjugate of (z₁ + z₂) = z̄₁ + z̄₂, and conjugate of (z₁ · z₂) = z̄₁ · z̄₂. Also (z̄)¯ = z.
Define a set and what is meant by the empty set.
A set is a well-defined collection of distinct objects (elements). The empty (null) set, denoted ∅ or { }, contains no elements.
Distinguish between a subset and a proper subset.
A ⊆ B means every element of A is in B. A is a proper subset (A ⊂ B) if A ⊆ B and A ≠ B (B has at least one element not in A).
If a set has n elements, how many subsets and how many proper subsets does it have?
It has 2ⁿ subsets and 2ⁿ − 1 proper subsets.

Show the other 32 Mathematics (Part I) - Algebra and Trigonometry flashcards

Define the union and intersection of two sets A and B.
A ∪ B = {x : x ∈ A or x ∈ B}; A ∩ B = {x : x ∈ A and x ∈ B}.
Define the difference A − B and the complement Aᶜ of a set.
A − B = {x : x ∈ A and x ∉ B}. The complement Aᶜ = U − A, where U is the universal set.
What are disjoint sets and overlapping sets?
Disjoint sets have no common elements (A ∩ B = ∅). Overlapping sets share at least one element (A ∩ B ≠ ∅).
State De Morgan's Laws for sets.
(A ∪ B)ᶜ = Aᶜ ∩ Bᶜ and (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ.
State the distributive laws for sets.
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
State the commutative and associative laws for union and intersection.
Commutative: A∪B=B∪A, A∩B=B∩A. Associative: (A∪B)∪C=A∪(B∪C), (A∩B)∩C=A∩(B∩C).
What is a Venn diagram used to represent?
A Venn diagram uses overlapping closed figures (usually circles) inside a rectangle (the universal set) to visually represent sets and the relationships/operations among them.
Define the Cartesian product A × B of two sets.
A × B = {(a, b) : a ∈ A and b ∈ B}, the set of all ordered pairs with first element from A and second from B. If |A|=m and |B|=n then |A×B|=mn.
Define a relation from set A to set B.
A relation from A to B is any subset of the Cartesian product A × B (a set of ordered pairs).
Define the domain and range of a relation.
The domain is the set of all first elements of the ordered pairs; the range is the set of all second elements.
Define a function from set A to set B.
A function f: A → B assigns to every element of A exactly one element of B (no element of A has two images and no element is left unmapped).
Distinguish between an into function and an onto (surjective) function.
In an into function, at least one element of the codomain has no pre-image (range ⊂ codomain). In an onto function the range equals the codomain (every element of B is an image).
Define a one-to-one (injective) function and a bijective function.
Injective: distinct elements of the domain have distinct images (f(a₁)=f(a₂) ⇒ a₁=a₂). Bijective: a function that is both one-to-one and onto.
Define a binary operation on a set S.
A binary operation * on S is a function that assigns to each ordered pair (a, b) of elements of S a unique element a*b that is also in S (closure).
List the four axioms a set with a binary operation must satisfy to be a group.
1) Closure, 2) Associativity, 3) Existence of an identity element e, 4) Existence of an inverse for every element. (An abelian group additionally satisfies commutativity.)
What is the difference between a semigroup, a monoid, and a group?
Semigroup: closure + associativity. Monoid: semigroup + identity element. Group: monoid + inverse for every element.
Define an abelian (commutative) group.
A group (G, *) in which the operation is commutative, i.e. a*b = b*a for all a, b ∈ G.
Define a matrix and its order (dimension).
A matrix is a rectangular array of numbers arranged in rows and columns. Its order is m × n where m is the number of rows and n the number of columns.
Define a square matrix, a row matrix and a column matrix.
Square matrix: equal number of rows and columns (n × n). Row matrix: a single row (1 × n). Column matrix: a single column (m × 1).
Define a diagonal matrix, a scalar matrix and an identity matrix.
Diagonal: square matrix with all off-diagonal entries zero. Scalar: diagonal matrix with all diagonal entries equal. Identity: scalar matrix with all diagonal entries 1.
Define the transpose of a matrix, and what is a symmetric matrix?
The transpose Aᵗ is obtained by interchanging rows and columns. A is symmetric if Aᵗ = A; it is skew-symmetric if Aᵗ = −A.
State the condition for two matrices to be conformable for multiplication and the order of the product.
A (m×n) and B (p×q) can be multiplied only if n = p (columns of A = rows of B); the product AB has order m × q.
Is matrix multiplication commutative? State the correct general property.
No, matrix multiplication is generally non-commutative: AB ≠ BA in general. However it is associative ((AB)C = A(BC)) and distributive over addition.
State the transpose properties (AB)ᵗ and (A+B)ᵗ.
(AB)ᵗ = Bᵗ Aᵗ (reversal law) and (A + B)ᵗ = Aᵗ + Bᵗ. Also (Aᵗ)ᵗ = A.
How do you evaluate the determinant of a 2×2 matrix [[a, b],[c, d]]?
det = ad − bc.
What is a singular matrix and a non-singular matrix?
A square matrix is singular if its determinant is 0; it is non-singular (invertible) if its determinant is non-zero.
State the determinant properties for det(Aᵗ) and det(AB).
det(Aᵗ) = det(A) and det(AB) = det(A) · det(B).
What happens to a determinant if two rows (or columns) are interchanged, or if a row is multiplied by a scalar k?
Interchanging two rows/columns changes the sign of the determinant. Multiplying one row/column by k multiplies the whole determinant by k. If two rows/columns are identical, the determinant is 0.
Give the formula for the inverse of a non-singular matrix A.
A⁻¹ = (1/det A) · adj(A), where adj(A) is the adjoint (transpose of the matrix of cofactors). This requires det A ≠ 0.
How do you find the inverse of a 2×2 matrix A = [[a, b],[c, d]]?
A⁻¹ = 1/(ad − bc) · [[d, −b],[−c, a]], provided ad − bc ≠ 0.
Using matrices, how is a system AX = B solved when A is non-singular?
By the matrix inverse method: X = A⁻¹B, valid when det(A) ≠ 0 (the system has a unique solution).
State Cramer's Rule for solving a system of linear equations.
For AX = B with det(A)=D≠0, each variable xᵢ = Dᵢ / D, where Dᵢ is the determinant of A with its i-th column replaced by the column B.

ICS (Intermediate in Computer Science) Mathematics (Part I) - Algebra and Trigonometry FAQ

What is in the ICS (Intermediate in Computer Science) Mathematics (Part I) - Algebra and Trigonometry syllabus?

ICS (Intermediate in Computer Science) Mathematics (Part I) - Algebra and Trigonometry is split into 12 chapters: Number Systems, Sets, Functions and Groups, Matrices and Determinants, Quadratic Equations, Partial Fractions, Sequences and Series, Permutation, Combination and Probability, Mathematical Induction and Binomial Theorem, Fundamentals of Trigonometry, Trigonometric Identities of Sum and Difference of Angles, Application of Trigonometry and Inverse Trigonometric Functions and Trigonometric Equations. Together they contain 44 topics.

How many chapters are in ICS (Intermediate in Computer Science) Mathematics (Part I) - Algebra and Trigonometry?

12 chapters, containing 44 topics. The largest chapter is Matrices and Determinants, with 5 topics.

Are there flashcards for ICS (Intermediate in Computer Science) Mathematics (Part I) - Algebra and Trigonometry?

Yes — a 52-card Mathematics (Part I) - Algebra and Trigonometry deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.